Is $\frac{0}{0}$ a set?

Question

Say we want to find a number $x$ such that: $$x^2-5x+6=0$$ So, there's no unique value of $x$ satisfying that but we still say $x={2,3}$

Is the situation with $\frac{0}{0}$ the same? I mean, finding the solution of $x\cdot 0=0$. And, since all values of $x$ are satisfying that so $x=1,1.5,1.29,\sqrt{2},e,\pi,5+6i...$ or $x=C$, the set of complex numbers.

$x=2,3$ is a shorthand for $x = 2$ or $x = 3$. Equivalently, the equation has the solution set ${2,3}$. For the equation $0x = 0$, there's no "or" form., but you can say that solution set is $\mathbb{C}$. — quasi, Mar 06 '17 at 09:29
But the notation $0/0$ refers to an operation between the operands $0$ and $0$. If it were defined, it would have to be a unique number. Of course, it's left undefined precisely because there's no good choice. — quasi, Mar 06 '17 at 09:33
The questions "Is $\frac{0}{0}$ a set?" and "Find the set of solutions $x$ in $\mathbb C$ of $x\cdot 0=0$" are not the same. The answer to the first question is: "No", because $\frac{0}{0}$ does not exist (in particular it is not a set). The answer to the second question is: $\mathbb C$. — Did, Mar 06 '17 at 09:39
You can define the symbol $\frac{0}{0}$ to be whatever you want, for example to be my name "freakish". The reason we don't do that is because whatever definition you put on $\frac{0}{0}$ it just doesn't fit with the established arithmetics. It becomes an isolated thing that we just don't want to deal with. From that point of view defining $\frac{0}{0}$ to be some set just doesn't work well with other parts of mathematics. — freakish, Mar 06 '17 at 10:02

score 0 · Answer 1 · 2017-03-06T10:58:54.510

0

Assuming that you formally defined the notation

$$\frac pq$$

to represent the solution set of the equation

$$q\cdot x=p,$$

just like

$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

would denote the solutions of

$$ax^2+bx+c=0^*$$ for any $a,b,c$, then yes, $\dfrac00=\mathbb C$.

Such a notation is never adopted. In particular, it conflicts with the well-established symbol $\dfrac00$ that designates an undeterminate form; an undeterminate form is no particular value but a way to state that the expression at hand must be evaluated as a limit (with a specific pattern).

$^*$In particular $(5\pm\sqrt{5^2-4\cdot1\cdot6})/(2\cdot1)$, equivalent to $(5\pm1)/2$. More generally,

$$\frac{p\pm q}r$$ would denote the solution set of

$$\frac r2x^2-px+\frac{p^2-q^2}{2r}=0$$ for $r\ne0$ and

$$\frac{p\pm q}0$$

would be undefined or undeterminate.

edited Mar 06 '17 at 10:58

answered Mar 06 '17 at 10:14

You could let fractions be defined as a set of solutions. But then arithmetic would get hairy. – Arthur Mar 06 '17 at 11:08
@Arthur: who does this comment address ? – Mar 06 '17 at 11:12
It's a PS to your "Such notation is never adopted" paragraph. – Arthur Mar 06 '17 at 11:22

score 0 · Answer 2 · answered Mar 06 '17 at 11:15

If you define $\frac{0}{0}$ as the set of $x$ such as $0x=0$ of course you will get all of R but the point about $\frac00$ as it arises in the calculus is to consider limits of fractions of quantities getting smaller and smaller. Such a situation is summarized by the phrase "indeterminate form of type $\frac00$". With that interpretation of $\frac00$ you can't define it as the set of $x$ such that $0x=0$ because that's imcompatible with the analysis of the concept of limit.

Is $\frac{0}{0}$ a set?

2 Answers2